How do I do axis-locking + edge/face intersection snapping? (with screenshot)
You can enable Snapping by pressing the magnet at the top of the screen. If you click on the drop-down menu next to the magnet icon, you can change what Blender will snap to.

To toggle snapping via a keyboard shortcut, press Shift+Tab. Pressing Ctrl+Shift+Tab will bring up a pop-up menu for changing the snapping object.

If you don't want to toggle snapping, you can just hold Ctrl instead to enable snapping only while Ctrl is held down.